We perform a comprehensive analysis of the set of parameters ${r_{i}}$ that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time $tau$, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between $tau$ and the energy spectrum and allowing the classification of ${r_{i}}$ into families organized in a 2-simplex, $delta^{2}$. Furthermore, the states determined by ${r_{i}}$ are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those $r_{i}$s in $delta^{2}$ correspondent to states whose orthogonality time is limited by the Mandelstam--Tamm bound from those restricted by the Margolus--Levitin one. Our results offer a complete characterization of the physical quantities that become relevant in both the preparation and study of the dynamics of three-level states evolving towards orthogonality.
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